Question

If the system of linear equations

Hint:

For infinitely many solutions, the coefficient matrix must be singular and the augmented system must be consistent.

Answer 1

Correct Answer: 3

Step 1: Force a singular coefficient matrix.
For infinitely many solutions the determinant of $\begin{bmatrix}1&1&1\\1&2&-1\\a&7&1\end{bmatrix}$ must vanish.

Step 2: Compute it.
The determinant is $15-3a$, so $15-3a=0$ gives $a=5$.

Step 3: Force consistency.
With $a=5$ the third row $(5,7,1)=3(1,1,1)+2(1,2,-1)$, so the right hand sides must obey the same combination.

Step 4: Solve for $b$.
That gives $b+1=3(1)+2b$, hence $b=-2$.

Step 5: Add up.
$a+b=5-2=3$.
\[ \boxed{3} \]